In this paper, we study traveling wave solutions of the chemotaxis systems \begin{ equation} \begin{ cases} u_{ t}=\Delta u -\chi_1\nabla( u\nabla v_1)+\chi_2 \nabla(u\nabla v_2 )+ u(a -b u), \qquad \ x\in\mathbb{ R} \\ \tau\partial_tv_1=(\Delta- \lambda_1 I)v_1+ \mu_1 u, \qquad \ x\in\mathbb{ R}, \\ \tau\partial v_2=(\Delta- \lambda_2 I)v_2+ \mu_2 u, \qquad \ \ x\in\mathbb{ R}, \end{ cases} (0.1) \end{ equation} where $\tau>0,\chi_{ i}> 0,\lambda_i> 0,\ \mu_i>0$ ($i=1,2$) and $\ a>0,\ b> 0$ are constants, and $N$ is a positive integer. Under some appropriate conditions on the parameters, we show that there exist two positive constant $ 0<c^{ *}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)<c^{ **}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)$ such that for every $c^{ *}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)\leq c<c^{ **}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)$, $(0.1)$ has a traveling wave solution $(u,v_1,v_2)(x,t)=(U,V_1,V_2)(x-ct)$ connecting $(\frac{ a}{ b},\frac{ a\mu_1}{ b\lambda_1},\frac{ a\mu_2}{ b\lambda_2})$ and $(0,0,0)$ satisfying $$ \lim_{ z\to \infty}\frac{ U(z)}{ e^{ -\mu z}}=1, $$ where $\mu\in (0,\sqrt a)$ is such that $c=c_\mu:=\mu+\frac{ a}{ \mu}$. Moreover, $$ \lim_{ (\chi_1,\chi_2)\to (0^+,0^+))}c^{ **}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)=\infty$$ and $$\lim_{ (\chi_1,\chi_2)\to (0^+,0^+))}c^{ *}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)= c_{ \tilde{ \mu}^*}, $$ where $\tilde{ \mu}^*={ \min\{ \sqrt{ a}, \sqrt{ \frac{ \lambda_1+\tau a}{ (1-\tau)_{ +}}},\sqrt{ \frac{ \lambda_2+\tau a}{ (1-\tau)_{ +}}}\}}$. We also show that $(0.1)$ has no traveling wave solution connecting $(\frac{ a}{ b},\frac{ a\mu_1}{ b\lambda_1},\frac{ a\mu_2}{ b\lambda_2})$ and $(0,0,0)$ with speed $c<2\sqrt{ a}$.
Categories: math.AP